COMPLETELY RIGHT INJECTIVE SEMIGROUPS By J. B. FOUNTAIN [Received 11 October 1972] Introduction Let 8 be a semigroup with 1. A right S-system is a set A together with a map A x 8 -> A satisfying x(st) = (xs)t and xl — x (x e A, s,t e S). We write A s to indicate that A is a right ^-system. An 8-homomorphism f: A s -> B s is a map /: A s -> B s such that f{xs) =f(x)s. A right 8- system A s is injective when any $-homomorphism C s -* vl^ can be extended to B s for any B s containing C s . We define a semigroup S with 1 to be completely right injective when every right $-system is injective. The study of such semigroups was initiated by Feller and Gantos in [3], [4], and [5] where characterizations of completely right injective semigroups which are unions of groups or inverse semigroups or both have been obtained. The main object of the present note is to find a characterization in the general situation. Feller and Gantos have shown in [3] that a necessary condition for a semigroup to be completely right injective is that every right ideal is generated by an idempotent. They have also given in [5] an example of a semigroup with 1 and 0 satisfying this condition but which is not completely right injective. We start by investigating semigroups in which every right ideal is generated by an idempotent. Using some results of T. E. Hall, we show that on such semigroups, Green's relation Jf is a right congruence. Turning to com- pletely right injective semigroups, we first find a necessary and sufficient condition which involves right ideals and right congruences for a semi- group to have the desired property. This enables us, by choosing appro- priate right congruences, to prove the desired characterization theorem. In the final section, we use the characterization to discover some elementary properties of completely right injective semigroups and to provide some new examples of such semigroups. We adopt the notation and terminology of Clifford and Preston ([2]). In addition, Green's relations &, ££', Jf on a semigroup 8 will be denoted by £% s ,J% s ,jtf' s when we want to emphasize the particular semigroup involved. Also, for each element a in any semigroup S, we define by V(a) = {x e S: axa — a and xax = x} the set of inverses of a in 8. Proc. London Math. Soc. (3) 28 (1974) 28-44